The space {1,2,3,4,5,6}. The random variable that takes

The Definition of probability:

is a branch of mathematics that is used in calculating likelihood of the
occurrence of a certain event. Probability is quantified as a number between 0
and 1. For example, if the probability of an event to occur is 0, thus, this is
expressing that the event is impossible to happen and if the probability of an
event to occur is 1, thus, this is expressing that the event will certainly

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Probability Theorems:

Theorem #1.  P(A) = 1 ? P(A’).

Theorem #2.  P(Ø) = 0.

#3.  If events A and B are such
that A ? B,
then P(A) ? P(B)

Theorem #4.  P(A) ? 1.

#5.  For any two events A and B, P(A ? B)
= P(A) + P(B) ? P(A ? B).


Types of Random Variables:

random variable is the variable whose all possible values are numerical outcomes
of a random phenomenon where only one variable is associated to each event in
the experiment.

an example:

tossing a coin the sample space will be {H,T}. These two outcomes might be
indicated by the variable X =1 if the outcome is H, X=0 if the outcome is T

a second example: if a dice is rolled and the face of the dice is seen or
observed. The sample space is {1,2,3,4,5,6}. The variable X here can represent
the number on the face which means here that it is equal to the sample space

random variable that takes countable number of values is called discrete.

an example of discrete variables:

can the X from the previous two examples.


Continuous random variables:

the name these random variables assume the value corresponding to any value in
an interval. As an example:

amount of the time (X) that a student takes to finish an exam (if the total
time of the exam is one hour) 0 ?
X ? 60.

Types of probability distributions:

The probability
distribution of a discrete random variable can be done as a table, graph, or
formula that will be specific the probability connected each possible value
that the random variable can be.

There are some
requirements that must be present as to distribute the discrete random variable

1. P(x) ? 0,
for all values of x.

2. ?p(x) = 1

Where the
summation of p(x) is over all possible values of x.

The mean: the sum of all the values divided by their number.

The variance of
a random variable:

 The standard deviation of a random variable:
the square root of the variance.

The Binomial

Many of the
experiments that we do or make could result in two possible outcomes as
Yes-No,Pass-Fail,etc.. As for example, when tossing a coin there are two
possible outcomes which are Tail or Heads. Random variables possessing these characteristics
are called binomial random variables.

These variables
have certain characteristics:

1. The
experiment consists of n identical trials.

2. There are
only two possible outcomes on each trial. We will denote one outcome by S (for
Success) and the other by F (for Failure).

3. The
probability of S remains the same from trial to trial. This probability is
denoted by p, and the probability of F is denoted by q = 1 ? p.

4. The trials
are independent.

5. The binomial
random variable X is the number of S’s in n trials.

The Poisson distribution:

When an event
happens randomly through time. For example, the occurring of hurricanes when
letting X be the number of times that the hurricanes occurring in an interval
of time. Then the following conditions must be accomplished in order to achieve
the Poisson distribution.

of a Poisson Random Variable 1.
The events occur at a constant average rate of ? per unit time.

2. Occurrences
are independent of one another.

3. More than
one occurrence cannot happen at the same time.

The probability
distributions for continuous random variables:

The probability
distribution of the continuous random variable is achieved by having it’s
density function. The following properties are achieved when the density
function F(X) with domain (a,b) :


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